EDO UNIVERSITY, IYAMHO EDO STATE, NIGERIA

Transcription

1 EDO UNIVERSITY, IYAMHO EDO STATE, NIGERIA MTH 122 :ELEMENTARY STATISTICS INTRODUCTION OF LECTURER Alhassan Charity Jumai is Lecturer of Mathematics at the Faculty of Physical Sciences, Edo University Iyamho, Edo State, Nigeria. She is a Mathematician (Registered) and a member of the Nigerian Mathematical Society (NMS). She is currently pursuing her PhD at University of Benin, Edo State. She has Master of Science (MSc) in Mathematics at the University of Agriculture Makurdi, Benue State and First degree (B.Sc) in Mathematics and computer Science at the University of Agriculture Makurdi, Benue State, Nigeria. Her research area include Mathematical Modelling and Simulations. Counting on the valuable experiences gained at workshop and seminars, she published journal articles extensively in the area of Modelling the Effects of Carriers and Treatment on the Transmission dynamics of typhoid fever and the Vertical Transmission and the Dynamics of HIV/AIDS in a Growing Population. GENERAL OVERVIEW OF LECTURES Statistical data: types, sources and methods of collection. Presentation of data: tables, charts and graphs. Frequency and cumulative distributions. Measures of location, partitions, dispersion, skewness and kurtosis. Concepts and principles of probability. Random Variables. Probability distributions: Binomial, Poisson. Regression and correlation: least squares estimation of simple linear regression, interpretation of regression coefficient; use of regression. The product-moment and rank correlation, their interpretations and uses. INTENDED LEARNING OUTCOMES At the end of this course, students are expected to: 1. Define Statistics, and state the types, sources and methods of collection 2.draw a frequency cumulative distribution table 3. Understand the Measures of location, partitions, dispersion, skewness and kurtosis

2 4. understand Concepts and principles of probability. Random Variables. Probability distributions: Binomial, Poisson 5. Solve problems on Regression and correlation: least squares estimation of simple linear regression, interpretation of regression coefficient; use of regression. Statistics. Statistics consists of a body of methods for collecting and analyzing data. From above, it should be clear that statistics is much more than just the tabulation of numbers and the graphical presentation of these tabulated numbers. Statistics is the science of gaining information from numerical and categorical data. Statistical methods can be used to find answers to the questions like: What kind and how much data need to be collected? How should we organize and summarize the data? How can we analyze the data and draw conclusions from it? How can we assess the strength of the conclusions and evaluate their uncertainty? That is, statistics provides methods for 1. Design: Planning and carrying out research studies. 2. Description: Summarizing and exploring data. 3. Inference: Making predictions and generalizing about phenomena represented by the data. Furthermore, statistics is the science of dealing with uncertain phenomenon and events. Statistics in practice is applied successfully to study the effectiveness of medical treatments, the reaction of consumers to television advertising, the attitudes of young people toward sex and marriage, and much more. It s safe to say that nowadays statistics is used in every field of science. Example 1.1 (Statistics in practice). Consider the following problems: agricultural problem: Is new grain seed or fertilizer more productive? medical problem: What is the right amount of dosage of drug to treatment? political science: How accurate are the gallups and opinion polls? economics: What will be the unemployment rate next year? technical problem: How to improve quality of

3 Descriptive and Inferential Statistics There are two major types of statistics. The branch of statistics devoted to the summarization and description of data is called descriptive statistics and the branch of statistics concerned with using sample data to make an inference about a population of data is called inferential statistics. Definition (Descriptive Statistics). Descriptive statistics consist of methods for organizing and summarizing information (Inferential Statistics). Inferential statistics consist of methods for drawing and measuring the reliability of conclusions about population based on information obtained from a sample of the population Descriptive statistics includes the construction of graphs, charts, and tables, and the calculation of various descriptive measures such as averages, measures of variation, and percentiles. In fact, the most part of this course deals with descriptive statistics Inferential statistics includes methods like point estimation, interval estimation and hypothesis testing which are all based on probability theory. Variables A characteristic that varies from one person or thing to another is called a variable, i.e, a variable is any characteristic that varies from one individual member of the population to another. Examples of variables for humans are height, weight, number of siblings, sex, marital status, and eye color. The first three of these variables yield numerical information (yield numerical measurements) and are examples of quantitative (or numerical) variables, last three yield non-numerical information (yield non-numerical measurements) and are examples of qualitative (or categorical) variables. Quantitative variables can be classified as either discrete or continuous. Discrete variables. Some variables, such as the numbers of children in family, the numbers of car accident on the certain road on different days, or the numbers of students taking basics of statistics course are the results of counting and thus these are discrete variables. Typically, a discrete variable is a variable whose possible values are some or all of the ordinary counting numbers like 0, 1, 2, 3,.... As a definition, we can say that a variable is discrete if it has only a countable number of distinct possible values. That is, a variable is is discrete if it can assume only a finite numbers of values or as many values as there are integers. Qualitative variable The number of observations that fall into particular class (or category) of the qualitative variable is called the frequency (or count) of that class. A table listing all classes and their frequencies is called a frequency distribution. In addition of the frequencies, we are often interested in the percentage of a class. We find the percentage by dividing the frequency of the class by the total number of observations and multiplying the result by 100. The percentage of the class, expressed as a decimal, is usually referred to as the relative frequency of the class.

4 Frequency in the class=relative frequency/total number of observation A table listing all classes and their relative frequencies is called a relative frequency distribution. The relative frequencies provide the most relevant information as to the pattern of the data. One should also state the sample size, which serves as an indicator of the creditability of the relative frequencies. Relative frequencies sum to 1 (100%). A cumulative frequency (cumulative relative frequency) is obtained by summing the frequencies (relative frequencies) of all classes up to the specific class. In a case of qualitative variables, cumulative frequencies makes sense only for ordinal variables, not for nominal variables. The qualitative data are presented graphically either as a pie chart or as a horizontal or vertical bar graph. A pie chart is a disk divided into pie-shaped pieces proportional to the relative frequencies of the classes. To obtain angle for any class, we multiply the relative frequencies by 360 degrees, which corresponds to the complete circle. A horizontal bar graph displays the classes on the horizontal axis and the frequencies (or relative frequencies) of the classes on the vertical axis. The frequency (or relative frequency) of each class is represented by vertical bar whose height is equal to the frequency (or relative frequency) of the class. In a bar graph, its bars do not touch each other. At vertical bar graph, the classes are displayed on the vertical axis and the frequencies of the classes on the horizontal axis. Nominal data is best displayed by pie chart and ordinal data by horizontal or vertical bar graph. Measure of central tendency Mean: The arithmetic mean of x of a set of n observations x is simply their average, Variable(x) Frequency(f) Product(xf) sum of the observations Mean= x = x number of observation n When calculating a mean from a frequency distribution, this becomes Mean=x = x x = n f For example, for the following frequency distribution, we need to add a third column showing the values of the product x x f, after which the mean can be found

5 So x = x f = = When calculating the mean from a frequency distribution with grouped data, the central value, Xm, of the class is taken as the x-value in forming the product xf. So, for the frequency distribution Variable(x) Frequency(f) n= f=30 and Xmf = 597 x = Xmf n =597 = Mode The mode of a set of data is that value of the variable that occurs most often. For instance, in the set of values 2,2,6,7,7,7,10,13,the mode is clearly 7.There could, of course, be more than one mode in asset of observations,e.g.23,25,25,25,27,27,28,28,28,has two modes,25 and 28,each of which appears three times Mode of a grouped frequency distribution The masses of 50 casting gave the following frequency distribution Mass x(kg) Frequency (f) If we draw the histogram, using the central values as the mid-points of the bases of the rectangles, we obtain Median of a set of data The third measure of central tendency is the median, which is the value of the middle term when all the observations are arranged in ascending or decending order. For example, with 4,7,8,9,12,15,26, the median is 9 Where there is an even number of values, the median is then the average of the two middle terms e.g 5,6,10,12,15,26,the median is then the average of the two middle terms e.g.5,6,10,12,14,17,23,30 has a median of = 13 2 Similarly for the set of values 13,4,18,23,9,16,18,10,20,6 the median is.. Rearranging the terms in order, we have 4,6,9,10,13,16,18,18,20, 23 and median= = 14.5 Median with grouped data Since the median is the value of the middle term, it divides the frequency histogram into two equal areas. This fact gives us a method for determing the median For example, the temperature of a component was monitored at regular intervals on 80 occasions. The frequency distribution was as follows: 2

6 Temperature (x 0 C) Frequency Measure of Dispersion Range: The difference between the highest and the lowest values in the set of observations. For example 26,27,28,29, 30. The range is 30-26=4 While that of 5,19,20,36, 60 the range is 60-5=55 Standard deviation The following rules are observed 1 the mean x of the set of n values is first calculated 2 The deviation of each of these values x 1, x 2, x 3, x n from the mean is calculated and the results squared i.e (x 1 x ) 2, (x 2 x ) 2, (x 3 x ) 2 (x n x ) 2 3 The average of the results is then found and the result is called the variance of set of observations (x x )2 n 4) The square root of the variance gives the standard deviation, denoted by the greek letter sigma σ Standard deviation σ = (x x )2 n So for the set of values 3,6, 7,8,11, 13 CONCEPT OF SKEWNESS The skewness of a distribution is defined as the lack of symmetry (i.e similar part not facing each other). It is a description of the tendency of a distribution to have a pronounced tail to the left (negative) or to the right (positive) extremes. The measures of skewness are General formula for skewness = (x i x )3 where s is the standard deviation ns 3 Karl Pearson Ist Coefficient of Skewness Sk, given by Mean - Mode Sk = s.d The sign of Sk gives the direction and its magnitude gives the extent of skewness If Sk > 0, the distribution is positively skewed, and if S, < 0 it is negatively skewed. So far we have seen that Sk is strategically dependent upon mode. If mode is not defined for a distribution we cannot find Sk. But empirical relation between mean, median and mode states that, for a moderately symmetrical distribution, we have Mean - Mode = 3 (Mean - Median) Hence Karl Pearson's 2 nd coefficient of skewness is defined in terms of median as Sk = 3(Mean - Mode) s.d

7 INTERPRETATION If skewness is zero, the data set is said to be symmetric or normally distributed. if the skewness is less than zero(negative),the data is said to be left skewed or negatively skewed, and if the skewed is greater than zero, the data set is said to be right skewed or positively skewed. Compute the Karl Pearson's ist coefficient of skewness from the following data: Height (in inches) Number of Persons Kurtosis is a measure of the degree of peakedness of flatness of a distribution Kurtosis= (x i x ) 4 ns 4 F) Coefficient of variation This is simply expressed as coefficient of variation= s 100 x 1 Example Calculate the range, standard deviation, variance and the coefficient of variation of the distribution from the data below Class Classmark F X-x Fx (X-x )

8 Concept of probability Probability is a measure of the likelihood that a particular event will occur in any one trial, or experiment, carried out in prescribed conditions. EXPERIMENT, OUTCOMES, AND SAMPLE SPACE An experiment is any operation or procedure whose outcomes cannot be predicted with certainty. The set of all possible outcomes for an experiment is called the sample space for the experiment. 1. If A is an event from a sample space S, then the probability of A, denoted as P(A),is calculated as total number of points in A P(A) = total number of points in S Example 1 If it is known that S={1,2,3,4,5,6},and that all outcomes are equally likely, then it follows that P(4)= 1, P(6)= 1 P(1) = P(even no) = 3 6 =1 2, P(odd no) = 3 6 =1 2 Sample space In a statistical experiment, the set of all possible occurrences or outcomes is known as the sample space. TREE DIAGRAMS AND THE COUNTING RULE In a tree diagram, each outcome of an experiment is represented as a branch of a geometric figure called a tree. EXAMPLE Figure 1 shows the sample spaces for three different experiments. For the experiment of tossing a coin twice, there are 4 outcomes. For the experiment of tossing a coin three times there are 8 outcomes. For the experiment of tossing a coin four times there are 16 outcomes. The outcomes are called branches of the tree and the device for showing the branches is called a tree diagram.

9 Examples on probability A bag contains balls that are identical in size and shape but 3 of them are red, 9 are white,6 are blue and 3 orange coloured. If a ball is picked at random, what is the probability that it is 1) red? Ii) White iii) blue? Iv) not orange Solution Total number of balls = =21 i) number of red balls P(red) = no of red balls total no of balls = 3 21 =1 7 ii) Number of white balls =9 Prob. of white ball = no of white balls iii) Number of blue balls = 6 no of blue balls total no of balls = 9 21 =3 7 P(Blue) = = 6 = 2 total no of balls 21 7 iv) No of orange balls=3 Prob(orange) = no of orange balls = 3 total no of balls 21 Prob (not orange) =1 1 7 = 6 7 Sampling with or without replacement This is concerned with the experiments that involved repetition or more than one trial. e.g Two balls are to be drawn one after the other from a bag containing 3 black,2 blue and 4 red balls. What is the probability that

10 a) if there is replacement, b) if there is no replacement i) both balls will be black ii) both balls will be blue iii) the first is blue and the second is red Solution If there is replacement total number=3+2+4=9 Probability that the first draw is black is 3 9 And probability that the second draw will be black is also 3 9 Hence, probability that both will be black is = 1 9 ii) Similarly, probability that both balls will be blue is = 4 81 iii) Probability that the first draw is blue is 2 9 And probability that the second is red is 4 9 prob. that 1st is blue and 2nd is red is = 8 81 b) If there is no replacement, Total number of balls = 9 i) Probability that first draw is black is 3 9,and that second draw is also black is 2 8 Solution If there is replacement total number=3+2+4=9 Probability that the first draw is black is 3 9 And probability that the second draw will be black is also 3 9 Hence, probability that both will be black is = 1 9 ii) Similarly, probability that both balls will be blue is = 4 81 iii) Probability that the first draw is blue is 2 9 And probability that the second is red is 4 9 prob. that 1st is blue and 2nd is red is = 8 81 b) If there is no replacement, Total number of balls = 9 i) Probability that first draw is black is 3 9, and that second draw is also black is 2 8 Hence, probability that both are black = = 1 12 ii) Probability that first is blue = 2 9 And probability that second is blue also = 1 8 Hence, prob. that both are blue = = 1 36 iii) Probability that first is blue = 2 9

11 And probability that second draw is red = 4 8 Introduction Many probability problems involve assigning probabilities to the outcomes of a probability experiment. These probabilities and the corresponding outcomes make up a probability distribution. There are many different probability distributions. One special probability distribution is called the binomial distribution. The binomial distribution has many uses such as in gambling, in inspecting parts, and in other areas. Discrete Probability Distributions In mathematics, a variable can assume different values. For example, if one records the temperature outside every hour for a 24-hour period, temperature is considered a variable since it assumes different values. Variables whose values are due to chance are called random variables. When a die is rolled, the value of the spots on the face up occurs by chance; hence, the number of spots on the face up on the die is considered to be a random variable. The outcomes of a die are 1, 2, 3, 4, 5, and 6, and the probability of each outcome occurring is 1/6. The outcomes and their corresponding probabilities can be written in a table, as shown, and make up what is called a probability distribution. Value,x Prob,p(x) A probability distribution consists of the values of a random variable and their corresponding probabilities. There are two kinds of probability distributions. They are discrete and continuous. A discrete variable has a countable number of values (countable means values of zero, one, two, three, etc.). For example, when four coins are tossed, the outcomes for the number of heads obtained are zero, one, two, three, and four. When a single die is rolled, the outcomes are one, two, three, four, five, and six. These are examples of discrete variables. A continuous variable has an infinite number of values between any two values. Continuous variables are measured. For example, temperature is a continuous variable since the variable can assume any value between 10 degrees and 20 degrees or any other two temperatures or values for that matter. Height and weight are continuous variables. Of course, we are limited by our measuring devices and values of continuous variables are usually rounded off. EXAMPLE: Construct a discrete probability distribution for the number of heads when three coins are tossed.

12 SOLUTION: Recall that the sample space for tossing three coins is TTT, TTH, THT, HTT, HHT, HTH, THH, and HHH. The outcomes can be arranged according to the number of heads, as shown. 0 heads TTT 1 head TTH, THT, HTT 2 heads THH, HTH, HHT 3 heads HHH Finally, the outcomes and corresponding probabilities can be written in a table, as shown. Outcome,x Probability, P(x) The sum of the probabilities of a probability distribution must be 1. A discrete probability distribution can also be shown graphically by labeling the x axis with the values of the outcomes and letting the values on the y axis represent the probabilities for the outcomes. The graph for the discrete probability distribution of the number of heads occurring when three coins are tossed is shown below. There are many kinds of discrete probability distributions; however, the distribution of the number of heads when three coins are tossed is a special kind of distribution called a binomial distribution. A binomial distribution is obtained from a probability experiment called a binomial experiment. The experiment must satisfy these conditions:

13 1. Each trial can have only two outcomes or outcomes that can be reduced to two outcomes. The outcomes are usually considered as a success or a failure. 2. There is a fixed number of trials. 3. The outcomes of each trial are independent of each other. 4. The probability of a success must remain the same for each trial. EXAMPLE: Explain why the probability experiment of tossing three coins is a binomial experiment. SOLUTION: In order to be a binomial experiment, the probability experiment must satisfy the four conditions explained previously. 1. There are only two outcomes for each trial, head and tail. Depending on the situation, either heads or tails can be defined as a success and the other as a failure. 2. There is a fixed number of trials. In this case, there are three trials since three coins are tossed or one coin is tossed three times. 3. The outcomes are independent since tossing one coin does not effect the outcome of the other two tosses. 4. The probability of a success (say heads) is 1/2 and it does not change Hence the experiment meets the conditions of a binomial experiment. Now consider rolling a die. Since there are six outcomes, it cannot be considered a binomial experiment. However, it can be made into a binomial experiment by considering the outcome of getting five spots (for example) a success and every other outcome a failure. In order to determine the probability of a success for a single trial of a probability experiment, the following formula can be used.

14 Notice that there were 3C2 or 3 ways to get two heads and a tail. The answer 3/8 is also the same as the answer obtained using classical probability that was shown in the first example. EXAMPLE: A die is rolled 3 times. Construct a probability distribution for the number of fives that will occur.

15 The Mean and Standard Deviation for a Binomial Distribution Suppose you roll a die many times and record the number of threes you obtain. Is it possible to predict ahead of time the average number of threes you will obtain? The answer is Yes. It is called expected value or the mean of a binomial distribution. This mean can be found by using the formula mean (µ)=np where n is the number of times the experiment is repeated and p is the probability of a success. The symbol for the mean is the Greek letter (mu). EXAMPLE: A die is tossed 180 times and the number of threes obtained is Recorded. Find the mean or expected number of threes.

16 Statisticians are not only interested in the average of the outcomes of a Probability experiment but also in how the results of a probability experiment vary from trial to trial. Suppose we roll a die 180 times and record the number of threes obtained. We know that we would expect to get about 30threes. Now what if the experiment was repeated again and again? In this case, the number of threes obtained each time would not always be 30 but would vary about the mean of 30. For example, we might get 28 threes onetime and 34 threes the next time, etc. How can this variability be explained? Statisticians use a measure called the standard deviation. When the standard deviation of a variable is large, the individual values of the variable are spread out from the mean of the distribution. When the standard deviation of a variable is small, the individual values of the variable are close to the mean. The formula for the standard deviation for a binomial distribution is standard deviation σ = np(1 p) The symbol for the standard deviation is the Greek letter σ (sigma).

17 EXAMPLE: An archer hits the bull s eye 80% of the time. If he shoots 100 Arrows, find the mean and standard deviation of the number of bull s eyes. If he travels to many tournaments, find the approximate range of values. SOLUTION: n=100, p=0.80, 1-p=1-0.80=0.20

18

19 Simple linear regression Is the study of dependence of one variable on another. Other types of regression analysis are non-linear regression, partial regression and multiple linear regression. Linear regression analysis is a tool for predicting the values of the dependent

20 variable from observed values of the independent variable. A linear equation relating Y on X is an equation of the form Y = a+ bx, Where a and b are constants. The graph of a linear equation is a straight line, a is the y intercept and b is the slope or gradient of the line. The scatter Diagram The scale for the independent variable lies along the X-axis and the scale of the dependent variable lies on the Y-axis. EXAMPLE 1 The scores of nine boys in JAMB mathematics examination were noted, before they then sat for another test. Both are shown in the table below. Construct a scatter diagram and draw the line of best fit. Are the two variables related Correlation coefficient The coefficient of correlation is another measure of closeness (relationship) between two sets of values. It is a study of the degree of interdependence between two variables. correlation can be positive, negative or zero.it values lies between -1 and 1 i.e -1 r 1 If r=0 there is no correlation If r=1 then, there is perfect correlation between X and Y If r=-1,perfect negative correlation If 0.5 r < 1, then, there is strong positive correlation We will consider two methods for calculating the coefficient of correlation. These are: a) The person s product moment correlation coefficient and b) The spearman;s rank correlation coefficient SUMMARY AND CONCLUSIONS Statistics consists of a body of methods for collecting and analyzing data. From above, it should be clear that statistics is much more than just the tabulation of numbers and the graphical presentation of these tabulated numbers. Statisticians are not only interested in the average of the outcomes of a Probability experiment but also in how the results of a probability experiment vary from trial to trial. Suppose we roll a die 180 times and record the number of threes obtained?

21 REFERENCES Johnson, R.A. & Bhattacharyya, G.K., Statistics: Principles and Methods, 2nd Edition. Wiley, Moore, D. & McCabe G., Introduction to the Practice of Statistics, 3th Edition. Freeman, Schaum s Outline Series New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Basics of statistics Jarkko Isotalo